Quirinius

Most sources claim he reigned around 6 AD but the Bible claims he was in reign also during the reign of Herod which ended approximately 4 BC. Is Luke 2:2 the verse that discredits Luke as well as Acts as Godly inspired sources? Therefore invalidating the Bible as we know it?

Part of Luke's birth narrative, Luke 2:2 reads "This was the first registration and was taken while Quirinius was governor of Syria." (ESV). This verse famously suggests a historical error on Luke's part in temporally locating the birth of Jesus. Quirinius' governorship did not overlap with Herod's reign (Herod died 4 BC and Quirinius became governor 6 AD) - so Jesus' birth could not happen under both.

But some clever interpreters keep inventing ways to reconcile Luke's narrative with the historical record. I wonder if the below solution is at all plausible. It argues that Quirinius may have been governor twice or even if he wasn't governor perhaps "he was governing in Syria. Luke uses the verb ἡγεμονεύω (hēgemoneuō)" - so this could refer to another position other than governor/legate specifically.

Some have suggested [Quirinius] may have been Legate of Syria twice. As we have seen, the Trivoli tombstone inscription is evidence this occurred. While critics have pointed out that Publius Quintus Varus was the Legate of Syria from 7-4 BC, there is some debate around who followed him as Legate in Syria. Holden and Geisler conclude, “The probability that Quirinius was governor of Syria on two different occasions also cannot be ignored – once while prosecuting military action against the Homonadensians between 12 and 2 BC, and then a second time beginning about 6 AD.”20

Even if he wasn’t the Legate of Syria, he may have held a different role that would be considered governing, consistent with Luke’s description. Jared M. Compton summarizes: “Quirinius’s personal chronology is not fully known, particularly around the years of Jesus’ birth. Thus, it is not impossible that he held another office at the time which Luke appropriately describes with (h[gemoneuontoj thj Suriaj) hegmoneuontos tēs Surias, a description as we saw which could also appropriately describe the office from which he took his well-known census.”21

- We have a fairly good record of the governors of Roman Syria, and thus there's little room for imagining an opening for a second Quirinius governorship overlapping with Herod.

- But I do wonder about this alternative translation of ἡγεμονεύω (hēgemoneuō) - could Luke have been referring to Quirinius being in some other role?

- Also assuming the above is plausible (say Quirinius was a sublegate of Syria in 6 AD), would a Roman census include census taking in Judea? At that time Judea was under the Herodian Kingdom, which later became a tetrarchy, and only in 6 AD became the Roman Province of Judea. Would client Kingdoms be involved in a Roman census? Or were only provinces involved in census taking?